Index is multiplicative for profinite groups
Suppose is a profinite group and are closed subgroups of with . Note that automatically becomes closed in . (Conversely, being closed in and being closed in would imply that is closed in ). Then, we have:
where denote the respective values for the index of a closed subgroup in a profinite group where the subgroup is and the group is . Similarly for (index of subgroup in group ) and (index of subgroup in group ).